64. The figure shows a fixed circle C with equation (x-1)+ y-1 and a shrinking circle C; with radiusr and center the...
Hypocycloid When a circle rolls on the inside of a fixed circle, any point P on the circumference of the rolling circle describes a hypocycloid. Let the fixed circle be x2 + y2 = a", let the radius of the rolling circle be b, and let the initial position of the tracing point P be A(a, 0). Find parametric equations for the hypocycloid, using as the parameter the angle θ from the positive x-axis to the line joining the circles'...
The figure below shows a curve C, parametrized by (a) The point P lies on C, and its r-coordinate is 4. Find the value of t at the point P according to the parametrization, and find the y-coordinate of P. equation in terms of r and y. line 4. as shown shaded in the figure. Find the area of R. (b) The line is normal to C at the point P. Express the line l using an (c) The bounded...
Recall that the upper half plane H ((x, y)ly > 0) gives a "model" for hyperbolic space. In this model, distance decreases as one moves up (i.e. the distance between (0, 1) and (1, 1) is 1, the distance between (0,2) and (1,2) is 1/2, and the distance between (0, y) and (1, y) is 1/y. Draw a picture to see that this is strange.) The geodesics on H are given by (1) half circles with center somewhere on the...
7. (10) Find the flaw in the following attempted proof of the parallel postulate by Wolfgang Bolyai (Hungarian, 1775 - 1856) (see Fig. 3). Given any point P not on a line l, construct a line 1' parallel to through P in the usual way: drop a perpendicular PQ to / and construct /" perpendicular to PQ. Let I" be any line through P distinct from l'. To see that /" intersects I, pick a point A on PQ between...
Figure shows a point P on the central axis through a thin ring (through the center of curvature and perpendicular to the plane of the ring). The ring holds a uniformly distributed charge Q = 5.00 x10-15 C and has radius R = 3.00 cm. Point P is at distance d= 25.0 cm from the ring's center. What is the potential V at P due to the ring's charge? - --- da
need help Find the length of the curve defined by the parametric equations y3In(t/4)2-1) from t 5 tot- 7 Find the length of parametized curve given by a(t) -0t3 -3t2 + 6t, y(t)1t3 +3t2+ 0t, where t goes from zero to one. Hint: The speed is a quadratic polynomial with integer coefficients. A curve with polar equation 14 7sin θ + 50 cos θ represents a line. Write this line in the given Cartesian form Note: Your answer should be...
Figure below shows a insulating rod having a uniformly distributed charge Q, the rod has been bent in a 120 degree circular arc of radius r. We place coordinate axes such that the axis of symmetry of the rod lies along the x axis and the origin is at the center of curvature P of the rod. In terms of Q and r, (a) what is the electric field due to the rod at point P (b) What is the...
y 1/(1-x) 5. The point P(2,-1) lies on the curve C If Q is the point (x,1/(1-x), use your calculator to find the slope of the secant line PQ a. (correct to six decimal places) for the following values of x: 1.5 (ii) 1.9 (i) 1.99 (iv) 1.999 (v) 2.5 (vi) 2.1 (vii) 2.01 (vii) 2.001 Using the results of part (a), guess the value of the slope of the tangent line to the curve at P(2,-1) i. b. Using...
The figure shows two charged particles on an x axis: -q = -6.40 × 10-19 C at x = -4.50 m and q = 6.40 × 10-19 C at x = 4.50 m. What is the x-component of the net electric field produced at point P at y = 4.00 m?
The figure shows two charged particles on an x axis: -q = -6.40 × 10-19 C at x = -2.00 m and q = 6.40 × 10-19 C at x = 2.00 m. What are the (a)x- and (b)y- components of the net electric field produced at point P at y = 3.00 m?